Finite Group Databasebeta

← #80187

Subgroups — #80187

Order 384.

416 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#946C211yesmaximal, characteristic
2D3×C22⋊Q8C211yesmaximal, characteristic
3C3×#451C211yesmaximal, characteristic, complemented
4#5228C211yesmaximal, characteristic, complemented
5#5188C212yesmaximal, complemented
6#5186C212yesmaximal, complemented
7#5044C211yesmaximal, characteristic, complemented
8#4813C212yesmaximal, complemented
9#4811C212yesmaximal, complemented
10D3×C22⋊C8C211yesmaximal, characteristic
11#4722C211yesmaximal, characteristic, complemented
12#3189—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×D3×Q8C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C2×C3×Q16C2212yescomplemented
17C3×C22⋊Q8C2211yescharacteristic
18#946—21no—
19#946—21no—
20#946C2212yes—
21#946C2211yescharacteristic
22#946C2212yes—
23C2×#216C2212yescomplemented
24C2×(C2×D3)⋊C4—21no—
25C2×C4⋊Dic3C2211yescharacteristic
26#924C2211yescharacteristic
27C2×Dic3⋊C4—21no—
28C2×D3×C8C2212yes—
29#897C2211yescharacteristic
30#896—21no—
31D3×C4⋊C4—21no—
32D3×C4⋊C4C2211yescharacteristic
33D3×C22⋊C4—21no—
34#878—21no—
35C3×#103C2212yescomplemented
36C3×C22⋊C8C2211yescharacteristic
37#835C2212yescomplemented
38#835C2212yescomplemented
39#830C2211yescharacteristic
40(C2×D3)⋊C8C2212yes—
41#818C2212yescomplemented
42#816C2212yescomplemented
43#807C2212yescomplemented
44C2×C22⋊Q8—31no—
45C2×C22⋊Q8—31no—
46#484—31no—
47#484—32no—
48#484—31no—
49#467—32no—
50#467—32no—
51#451—31no—
52#451—32no—
53#451D311yescharacteristic, complemented
54C2×C22⋊C8—31no—
55C23×D3D411yescharacteristic
56C2×C3×Q8—21no—
57C2×C3×Q8—42no—
58C2×C3×Q8C2311yescharacteristic
59C2×C3×Q8C2312yes—
60C22×C3×C4—21no—
61C22×C3×C4C2311yescharacteristic
62C22×Dic3—21no—
63C22×Dic3D411yescharacteristic
64D3×Q8—42no—
65C2×C4×D3—21no—
66C2×C4×D3—21no—
67C2×C4×D3—42no—
68C2×C4×D3D411yescharacteristic
69C2×C4×D3D411yescharacteristic
70C2×C4×D3C2311yescharacteristic
71C2×C4×D3C2311yescharacteristic
72C2×Dic6—21no—
73C2×Dic6C2311yescharacteristic
74C3×Q16—42no—
75C2×C3×C8C2312yes—
76C3×C4⋊C4—21no—
77C3×C4⋊C4C2311yescharacteristic
78C3×C22⋊C4—21no—
79#217—21no—
80#216—42no—
81(C2×D3)⋊C4—21no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4—22no—
85C4⋊Dic3—21no—
86C4⋊Dic3—21no—
87C4⋊Dic3C2312yes—
88C4⋊Dic3C2311yescharacteristic
89Dic3⋊C4—21no—
90Dic3⋊C4—21no—
91Dic3⋊C4—21no—
92Dic3⋊C4—22no—
93Dic3⋊C4—22no—
94C2×C3⋊C8C2312yes—
95D3×C8—42no—
96C22×Q8—31no—
97C22×Q8C2×D311yescharacteristic
98C23×C4—31no—
99C2×Q16—32no—
100C2×Q16C2×D312yescomplemented
101C22×C8—32no—
102C22⋊Q8—32no—
103C22⋊Q8—31no—
104C22⋊Q8—32no—
105C22⋊Q8—31no—
106C22⋊Q8—31no—
107C22⋊Q8—61no—
108C22⋊Q8—61no—
109C22⋊Q8—61no—
110C22⋊Q8—61no—
111C22⋊Q8C2×D311yescharacteristic
112C2×C4⋊C4—31no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—61no—
115C2×C4⋊C4—61no—
116C2×C22⋊C4—61no—
117C2×C22⋊C4—61no—
118#107—32no—
119#107—32no—
120#103—32no—
121#103—32no—
122#103—32no—
123#103C2×D312yescomplemented
124C22⋊C8—31no—
125C22⋊C8—32no—
126C22⋊C8C2×D311yescharacteristic
127C23×C3C2×D411yescharacteristic
128C22×D3—41no—
129C22×D3—41no—
130C22×D3Q1614yescomplemented
131C22×D3C2×D411yescharacteristic
132C22×D3C2×D411yescharacteristic
133C3×Q8—22no—
134C3×Q8—22no—
135C3×Q8—22no—
136C3×Q8—42no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—42no—
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2411yescharacteristic
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—21no—
147C2×Dic3—41no—
148C2×Dic3—42no—
149C2×Dic3C2×D411yescharacteristic
150C2×Dic3C2×D411yescharacteristic
151C4×D3—22no—
152C4×D3—22no—
153C4×D3—41no—
154C4×D3—41no—
155C4×D3—42no—
156C4×D3—42no—
157Dic6—22no—
158Dic6—42no—
159C3×C8—22no—
160C3⋊C8—22no—
161C24—31no—
162C2×Q8—21no—
163C2×Q8—31no—
164C2×Q8—42no—
165C2×Q8—61no—
166C2×Q8—122no—
167C2×Q8C22×D311yescharacteristic
168C2×Q8C22×D312yes—
169C22×C4—21no—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—31no—
174C22×C4—31no—
175C22×C4—61no—
176C22×C4—61no—
177C22×C4—61no—
178C22×C4—122no—
179C22×C4C22×D311yescharacteristic
180Q16—42no—
181Q16—122no—
182C2×C8—32no—
183C2×C8—122no—
184C2×C8C22×D312yes—
185C4⋊C4—21no—
186C4⋊C4—31no—
187C4⋊C4—32no—
188C4⋊C4—61no—
189C4⋊C4—61no—
190C4⋊C4—62no—
191C4⋊C4—62no—
192C4⋊C4—61no—
193C4⋊C4—61no—
194C4⋊C4—61no—
195C4⋊C4C22×D311yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—61no—
198C22⋊C4—61no—
199C22⋊C4—62no—
200C22⋊C4—62no—
201C22⋊C4—61no—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3C2×Q1612yes—
205C22×C3C22×D411yescharacteristic
206C2×D3—21no—
207C2×D3—21no—
208C2×D3—21no—
209C2×D3—21no—
210C2×D3—44no—
211C2×D3C2×Q1612yes—
212C2×D3C2×Q1612yes—
213C3×C4—21no—
214C3×C4—41no—
215C3×C4—42no—
216C3×C4—41no—
217C3×C4H(5,𝔽2)12yes—
218Dic3—21no—
219Dic3—21no—
220Dic3—41no—
221Dic3—41no—
222Dic3—42no—
223C23—31no—
224C23—31no—
225C23—34no—
226C23—121no—
227C23—121no—
228C23D3×D411yescharacteristic
229Q8—22no—
230Q8—22no—
231Q8—22no—
232Q8—42no—
233Q8—62no—
234Q8—122no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—31no—
240C2×C4—31no—
241C2×C4—42no—
242C2×C4—62no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4—121no—
248C2×C4—121no—
249C2×C4—122no—
250C2×C4—122no—
251C2×C4—121no—
252C2×C4—122no—
253C2×C4D3×D411yescharacteristic
254C2×C4C23×D311yescharacteristic
255C8—22no—
256C8—62no—
257C2×C3—22no—
258C2×C3#53411yescharacteristic
259C2×C3C22×Q1611yescharacteristic
260C2×C3#57511yescharacteristic
261D3—22no—
262D3—22no—
263C22—21no—
264C22—21no—
265C22—32no—
266C22—32no—
267C22—61no—
268C22—61no—
269C22—61no—
270C22—61no—
271C22—124no—
272C22D3×Q1612yes—
273C22C2×D3×D411yescharacteristic
274C4—21no—
275C4—41no—
276C4—42no—
277C4—41no—
278C4—61no—
279C4—61no—
280C4—121no—
281C4—122no—
282C4—121no—
283C4#100412yes—
284C3#318911yescharacteristic, complemented, Sylow
285C2—22no—
286C2—62no—
287C2—62no—
288C2#558911yescharacteristic
289C2C2×D3×Q1611yescharacteristic
290C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
prime radicalC23×D3D4
hypercenter#451D3
hypocenterC3#3189
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C3×#451C2semidirect
4. #5228C2semidirect
5. #5188C2semidirect
6. #5186C2semidirect
7. #5044C2semidirect
8. #4813C2semidirect
9. #4811C2semidirect
11. #4722C2semidirect
16. C2×C3×Q16C22semidirect
23. C2×#216C22semidirect
35. C3×#103C22semidirect
37. #835C22semidirect
38. #835C22semidirect
41. #818C22semidirect
42. #816C22semidirect
43. #807C22semidirect
53. #451D3semidirect
100. C2×Q16C2×D3semidirect
123. #103C2×D3semidirect
130. C22×D3Q16semidirect
284. C3#3189semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#9462
D3×C22⋊Q82
C3×#4512
#52282
#51882
#51862
#50442
#48132
#48112
D3×C22⋊C82
#47222
#31893